Some quantitative results on Lipschitz inverse and implicit functions theorems

Abstract

Let $ f: \mathbb{R} ^ n \rightarrow \mathbb{R}^n $ be a Lipschitz mapping with generalized Jacobian at $x_0$, denoted by $\partial f(x_0)$, is of maximal rank. F. H. Clarke (1976) proved that $f$ is locally invertible. In this paper, we give some quantitative assessments for Clarke's theorem on the Lipschitz inverse, and prove that the class of such mappings are open. Moreover, we also present a quantitative form for Lipschitz implicit function theorem.